reserve Al for QC-alphabet;
reserve a,b,c,d for object,
  i,k,n for Nat,
  p,q for Element of CQC-WFF(Al),
  x,y,y1 for bound_QC-variable of Al,
  A for non empty set,
  J for interpretation of Al,A,
  v,w for Element of Valuations_in(Al,A),
  f,g for Function,
  P,P9 for QC-pred_symbol of k,Al,
  ll,ll9 for CQC-variable_list of k,Al,
  l1 for FinSequence of QC-variables(Al),
  Sub,Sub9,Sub1 for CQC_Substitution of Al,
  S,S9,S1,S2 for Element of CQC-Sub-WFF(Al),
  s for QC-symbol of Al;
reserve vS,vS1,vS2 for Val_Sub of A,Al;
reserve B for Element of [:QC-Sub-WFF(Al),bound_QC-variables(Al):],
  SQ for second_Q_comp of B;
reserve B for CQC-WFF-like Element of [:QC-Sub-WFF(Al),
  bound_QC-variables(Al):],
  xSQ for second_Q_comp of [S,x],
  SQ for second_Q_comp of B;

theorem Th34:
  [:QC-WFF(Al),vSUB(Al):] c= dom QSub(Al)
proof
  let a be object;
  assume a in [:QC-WFF(Al),vSUB(Al):];
  then consider b,c being object such that
A1: b in QC-WFF(Al) and
A2: c in vSUB(Al) and
A3: a = [b,c] by ZFMISC_1:def 2;
  reconsider Sub = c as CQC_Substitution of Al by A2;
  reconsider p = b as Element of QC-WFF(Al) by A1;
A4: now
    set b = {};
    set a = [[p,Sub],b];
    assume not p is universal;
    then p,Sub PQSub b by SUBSTUT1:def 14;
    then a in QSub(Al) by SUBSTUT1:def 15;
    hence thesis by A3,FUNCT_1:1;
  end;
  now
    set b = ExpandSub(bound_in p,the_scope_of p, RestrictSub(bound_in p,p,Sub)
    );
    set a = [[p,Sub],b];
    assume p is universal;
    then p,Sub PQSub b by SUBSTUT1:def 14;
    then a in QSub(Al) by SUBSTUT1:def 15;
    hence thesis by A3,FUNCT_1:1;
  end;
  hence thesis by A4;
end;
